Elliptically Contoured Models in Statistics and Portfolio Theory PDF
Preview Elliptically Contoured Models in Statistics and Portfolio Theory
Arjun K. Gupta Tamas Varga Taras Bodnar Elliptically Contoured Models in Statistics and Portfolio Theory Second Edition Elliptically Contoured Models in Statistics and Portfolio Theory Arjun K. Gupta • Tamas Varga (cid:129) Taras Bodnar Elliptically Contoured Models in Statistics and Portfolio Theory Second Edition 123 ArjunK.Gupta TamasVarga DepartmentofMathematicsandStatistics Damjanich,Budapest BowlingGreenStateUniversity Hungary OH,USA TarasBodnar DepartmentofMathematics Humboldt-UniversityofBerlin Berlin,Germany ISBN978-1-4614-8153-9 ISBN978-1-4614-8154-6(eBook) DOI10.1007/978-1-4614-8154-6 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013945413 ©SpringerScience+BusinessMediaNewYork1993,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Dedicatedtothememoryofmyparentsthe lateSmt.LeelaandSh.AmarnathGupta. (AKG) DedicatedtomychildrenTere´zia,Julianna, andKristo´f. (TV) DedicatedtomywifeOlhaandtomy childrenBohdanandAnna-Yaroslava. (TB) Preface Inmultivariatestatisticalanalysis,ellipticaldistributionshaverecentlyprovidedan alternativetothenormalmodel.Mostofthework,however,isspreadoutinjournals throughout the world and is not easily accessible to the investigators. Fang, Kotz, andNgpresentedasystematicstudyofmultivariateellipticaldistributions;however, they did not discuss the matrix variate case. Fang and Zhang have summarized the results of generalized multivariate analysis which include vector as well as the matrix variate distributions. On the other hand, Fang and Anderson collected research papers on matrix variate elliptical distributions, many of them published for the first time in English. They published very rich material on the topic, but the results are given in paper form which does not provide a unified treatment of the theory. Therefore, it seemed appropriate to collect the most important results on the theory of matrix variate elliptically contoured distributions available in the literatureandorganizetheminaunifiedmannerthatcanserveasanintroductionto thesubject. The book will be useful for researchers, teachers, and graduate students in statistics and related fields whose interests involve multivariate statistical analysis anditsapplicationintoportfoliotheory.PartsofthisbookwerepresentedbyArjun K.GuptaasaonesemestercourseatBowlingGreenStateUniversity.Knowledgeof matrixalgebraandstatisticsatthelevelofAndersonisassumed.However,Chap.1 summarizessomeresultsofmatrixalgebra.Thischapteralsocontainsabriefreview oftheliteratureandalistofmathematicalsymbolsusedinthebook. Chapter 2 gives the basic properties of the matrix variate elliptically contoured distributions, such as the probability density function and expected values. It also presentsoneofthemostimportanttoolsofthetheoryofellipticaldistributions,the stochasticrepresentation. Theprobabilitydensityfunctionandexpectedvaluesareinvestigatedindetailin Chap.3. Chapter4focusesonellipticallycontoureddistributionsthatcanberepresented asmixturesofnormaldistributions. The distributions of functions of random matrices with elliptically contoured distributionsarediscussedinChap.5.Specialattentionisgiventoquadraticforms. vii viii Preface CharacterizationresultsaregiveninChap.6. Thenextthreechaptersaredevotedtostatisticalinference.Chapter7focuseson estimationresults,whereasChap.8isconcernedwithhypothesistestingproblems. InferenceforlinearmodelsisstudiedinChap.9. Chapter 10 deals with the application of the elliptically contoured distributions for modeling financial data. We present distributional properties of the estimated maincharacteristicsofoptimalportfolios,likevarianceandexpectedreturnassum- ingthattheassetreturnsareellipticallycontoureddistributed.Thejointdistributions of the estimated parameters of the efficient frontier are derived as well as we provideexactinferenceproceduresforthecorrespondingpopulationvalues.Wealso studythedistributionalpropertiesoftheestimatedweightsoftheglobalminimum varianceportfolioindetail. In Chap.11, we consider a further extension of matrix variate elliptically contoured distributions that allows us to model the asymmetry in data. Here, first themultivariateskewnormaldistributionispresentedanditsmatrixgeneralization isdiscussed.Wealsostudythemainpropertiesofthisdistribution,likemoments,the densityfunction,andthemoment-generatingfunction.Next,theskewt-distribution is introduced as well as the general class of matrix variate skew elliptically con- toureddistributions.Moreover,wepresentthedistributionalpropertiesofquadratic formsinskewellipticaldistributionsanddiscusstheinferenceprocedures.Anappli- cationintoportfoliotheoryisdiscussedaswell.Finally,anup-to-datebibliography hasbeenprovided,alongwithauthorandsubjectindexes.Thematerialsinthefirst ninechaptersarefromthebookEllipticallyContouredModelsinStatisticsbythe firsttwoauthors.ThematerialinChaps.10and11istakenfromthepapersofthe authors. Permission of their publishers Kluwer Academic Publishers (http://www. wkap.com), Japan Statistical Society (http://www.jss.gr.jp), Springer (http://www. springer.com), and Taylor and Francis (http://www.tandfonline.com/) is gratefully acknowledged. WewouldliketothanktheDepartmentofMathematicsandStatistics,Bowling GreenStateUniversity,andtheDepartmentofMathematics,HumboldtUniversity ofBerlin,forsupportingourendeavorandforprovidingthenecessaryfacilitiesto accomplish the task. The first author is thankful to the Biostatistics Department, UniversityofMichigan,forprovidinghimtheopportunitytoorganizethematerial in its final form. Thanks are also due to Professors D. K. Nagar, M. Siotani, J. Tang, and N. Nguyen for many helpful discussions. He would also like to acknowledge his wife, Meera, and his children, Alka, Mita, and Nisha, for their support throughout the writing of the book. The second author is thankful to his mother Edit for her support in the early stages of the work on this book. The third author acknowledges the support of the Department of Statistics, European UniversityViadrinaandtheGermanResearchFoundation(DFG)viatheResearch Unit 1735 “Structural Inference in Statistics: Adaptation and Efficiency”. Thanks are also due to Professors W. Schmid and Y. Yelejko. He is also greatly thankful to his wife Olha and to his children Bohdan and Anna-Yaroslava for providing considerablehelpduringthepreparationofthebook. Preface ix FinallyoursincerethankstoMarcStrauss,Springer,forhishelpateverystage ofthecompletionofthisbook. BowlingGreen,USA ArjunK.Gupta Budapest,Hungary TamasVarga Berlin,Germany TarasBodnar
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